1. 23 - 1 Module 23: Proportions: Confidence Intervals and Hypothesis Tests, Two Samples This module examines confidence intervals and hypothesis test for two independent random samples for proportions. Reviewed 06 June 05 /MODULE 23

2. 23 - 2 P 1 = Parameter for population one P 2 = Parameter for population two x 1 = Number in sample one with the characteristic x 2 = Number in sample two with the characteristic n 1 = Total number in sample one n 2 = Total number in sample two p 1 = x 1 / n 1, estimate for sample one p 2 = x 2 / n 2, estimate for sample two Proportions: Two Independent Random Samples

3. 23 - 3 Hypothesis Test: Proportions from two samples H 0 : P 1 = P 2 vs. H 1 : P 1  P 2 p 2 = x 2 /n 2, p 1 = x 1 /n 1 The test is based on:

4. 23 - 4 Example: AJPH, Nov. 1977;67:1033 - 1036

5. 23 - 5

6. 23 - 6 Proportions for Two Independent samples 1. The hypothesis: H 0 : P M = P F vs. H 1 : P M  P F 2. The assumptions: Independent random samples 3. The  -level:  = 0.05 4. The test statistic: 5. The rejection region: Reject if z not between  1.96

7. 23 - 7 6. The test result: 7. The conclusion:Accept H 0 : P M = P F, since z is between ±1.96

8. 23 - 8 Confidence Interval for P M - P F  = 0.05

9. 23 - 9

10. 23 - 10 Example: Two Vaccines In testing two new vaccines, for one group 137 of 200 persons became infected. For the second group, 98 of 150 became ill. Test the hypothesis that the two vaccines are equally effective. 1. The hypothesis: H 0 : P 1 = P 2 vs. H 1 : P 1  P 2 2. The assumptions: Independent random samples 3. The  -level :  = 0.05 4. The test statistic:

11. 23 - 11 5. The rejection region: Reject H 0 : P 1 = P 2, if z is not between ±1.96 6. The result: 7. The conclusion: Accept H 0 : P 1 = P 2, since z is between ±1.96

12. 23 - 12 Example: AJPH, Sept. 1998;88:1319 - 1324

13. 23 - 13

14. 23 - 14 Hypothesis test for proportion of Asthma or Wheezy Bronchitis in Social Class I vs. Social Class II n I = 191 n II = 1,173 p I = 0.047 p II = 0.078 1. The hypothesis: H 0 : P I = P II vs. H 1 : P I  P II 2. The assumptions: Independent Random Samples Binomial Data 3. The  -level :  = 0.05 4. The test statistic: Hypothesis Test

15. 23 - 15 5. The rejection region: Reject H 0 : P I = P II, if z is not between ±1.96 6. The result: 7. The conclusion: Accept H 0 : P I = P II, since z is between ± 1.96

16. 23 - 16 n I = 191 n II = 1,173 p I = 0.047 p II = 0.078 Confidence Interval for P I - P II

17. 23 - 17